There is much need of tables of mathematical results which are not numerical ; such as the following, the only ones of which we arc able to speak. The first, the well-known collection of indefinite integrals by Meier Hirsch,' Integral Midi]; Berlin, 1810, of which an English edition was published in 1523, Svc. The second, a table of definite integrals, with reference to their sources, by Bierens de Haan,' Tables d'Integrales dailies; being the fourth volume of the ' Amsterdam Transactions,' Amsterdam, 1858.
As to astronomical tables, it would be impossible to give any account of the enormous mass which exists or has existed ; nor would such an account be of auy use, except for astronomical history. They may be divided into two classes : first, the tables of observations published by public or private observatories ; secondly, the fundamental tables deduced from observations, to aid in the deduction of future predic tions. As to the former, every well-conducted observatory in full work publishes periodically (at intervals of one or two years) its volume of observations, latterly with their reductions. As to the second class, they are not the daily materials of the astronomer, but of the com puter of his ephemeris, who supplies the necessary predictious for the current year. in England the Nautical Almanac gives in the preface full references to the tables employed in predicting places, whether of sun, moon, planets, or stars. For general purposes connected with the elements of the solar system, sec Bally,' Astroiffimical Tables and Formulze,' London, 1S27. The most complete list of the elements of the solar system recently published is at the end of Dr. Mitchel's Popular Astronomy' (U.S.), and also of Mr. Tomlinson's English edition of the same work.
The tables in the other physical sciences are mostly collections of facts, and, we believe, generally speaking, by no means so complete as they might be. The value of tabular information seems to be not sufficiently felt. A large portion of every book of chemistry, for instance, is a detailed statement in words at length of facts which might with great advantage be made the components of a table.
§ 8. It remains to speak of commercial tables, a subject of great interest in this country, which has produced a great many. The
mathematical tables connected with this subject may be divided into those intended to facilitate calculations of money with regard to other countries, aud with regard to transactions in this country; to which we must add, as distinct beads, tables of annuities and other life con tingencies, and metrological tables, or tables of weights and measures. Of all these we shall only mention a very few.
The most complete work on foreign exchanges, and on the weights and measures of England, as compared with those of other countries, is The Universal Cambiat,' &c., London, 1821 (2nd edition), 2 vols. 4to. (with supplements), by the late Dr. Patrick Kelly. We may also mention Tiarks's 'Arbitration of Exchanges,' London, 1817.
Tables of interest of money begin with Stevinus, who in the 'Practique d'Arithmetique; appended to his Arithmetic, Leyden, 1,585, reprinted by Albert Girard in Stevinus's collected works, 1626, gave the first tables of compound interest and annuities. They precede the famous tract La Disme,' in which decimal fractious were first proposed. And as this Praetique should rather have been at the beginning than at the end, if rational arrangement had been studied ; and as the ' again should have preceded it, on the same supposition ; we must infer it to be most likely that the tracts were placed in the order in which they were written. If this be the case, then it is pretty certain that these tables of compound interest suggested decimal fractions, the account of which speedily follows them. They are constructed as follows :—Ten millions being taken as the base (or root, as Stcvinus calls it), and a rate, say five per cent., being chosen, the present value of ten millions due at the end of 1, 2, &e., up to 30 years, are put in a column, to the nearest integer. By their sides are the sums of their values, which give the present values of the several annuities of ten million, as follows :— The rates arc from 1 to 16 per cent., and also for 1 in 15, 1 in 16, &c., to I in 22; or, as the French say, denier quinze, denier seize, etc. At the end is a direction to dispense, when convenient, with some of the last, figures.